The festival season is upon us. Between now and the end of August, you could pretty much whirl from one theatre fest to another. Next month alone, there's the Brighton festival and fringe, Mayfest in Bristol, the Norfolk and Norwich, Pulse in Ipswich and the new Incoming festival in London, curated by A Younger Theatre, which looks exciting.

Nobody, me least of all, is going to complain about the proliferation of theatre festivals, but as the number gets ever bigger, it's good to think hard about what purpose a festival serves, who it serves and what legacy it leaves for artists and audiences.

On Wednesday, I was in Glasgow for the first of the five days of the Buzzcut festival. This year's edition – the third – is being held at the Pearce Institute, a community centre in Govan about 15 minutes on the subway from the centre of town. It's almost certainly the first time that I've ever been at a live-art festival where the audience extends beyond the usual suspects. At one point there were OAPs watching a piece of pretty hardcore live art and children sitting in the cafe being led by artists in a discussion about a show – not a children's show – that they'd just seen. Too often, performance art is simply dismissed as silly or talked about only in an alienating high-flown academic language that puts most people off. Buzzcut suggests that, presented in the right context, a great deal of this work can be rewarding for everyone.

The range of performances on offer is terrific. I found myself in a nest listening to birdsong in Amy Rosa's one-on-one performance And You Will Find Yourself Again Only in the Simple and Forgotten Things. I enjoyed Louise Ahl's send-up of the therapy industry, NICE Healing, and was fascinated to watch Tom Frankland's durational doodle emerge as he sat in a cafe and spoke to people. I was bowled over by Lucy Hutson's Britney Spears Custody Battle vs Zeus in Swan Rape Shocker, a provoking and very funny piece that considers what we believe in and just how hard it is to lead an ethical life. I stood in the nearby graveyard and thought about the dead and my relationship to them in Edward Crawley's 5 Minutes to Move Me, and took A Spoonful of something sweet in Lou Brodie and Tashi Gore's installation.

Brian Lobel is performing You Have to Forgive Me, You Have to Forgive Me, You Have to Forgive Me at Buzzcut

The things that mark out Buzzcut are that it is artist-led, it is completely free, it is exceptionally welcoming and friendly, and it puts accessibility very high on its agenda – necessary in Govan, where many would not have the money to spend on art. Alongside the festival, graduates of the Royal Conservatoire of Scotland's influential Contemporary Performance Practice course are creating a new young people's company for the area. Nic Green (who made the brilliant Trilogy) is working on a show about the history of Govan and leading walks around the area during the festival too.

This is a festival founded on a generosity of spirit and an understanding that it is the quality of experience for artists and audiences that matters. Rosana Cade, who co-founded the festival in 2012 with Nick Anderson, talks about being invited to other festivals, often as a paid performer, but being isolated in digs or hotels and never even meeting the person who booked her show in the first place. Buzzcut may not pay its artists, but it does really look after them, accommodating them with other artists, feeding them and really taking note of what they are doing. Cade and Anderson aim to see every single one of the 50 performances they have programmed.

Artists respond to this context. Established practitioners such as Action Hero and Brian Lobel, who can certainly command fees for their work, are happy to take part in a festival that looks after them and sets itself up in such a way that those starting out on their careers meet on equal terms with those who have made reputations for themselves. Like that other artist-led initiative, Forest Fringe, Buzzcut is breaking the mould of how a festival can and should operate. It's on until Sunday. Next year, I hope I'll be staying longer.